C₄⋊#272 #6231

order	216
isoclinism	isoclinic class of #6229
automorphism	Aut(G) = D₄×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₂×C₃⋊D₃ · Out(G) = C₂×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

Fitting	C₄×H(3,𝔽₃) ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ D₄

88 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

(2,5,6)(4,8,9)(10,11,12,13);(2,4)(5,9)(6,8)(11,13);(1,2,4)(3,6,8)(5,9,7)

Polycyclic presentation:

2^3,3^3|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 68);